The equation $3 x^{2} - 12 x + (n - 5) = 0$ has equal roots. Find the value of n.

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Solution

Step: Finding the value of n.
Given,
$3 x^{2} - 12 x + (n - 5) = 0$
On comparing the given equation with $a x^{2} + b x + c = 0$ we get
$a = 3$
$b = - 12$
$c = n - 5$
$D = b^{2} - 4 a c$
$= (- 12)^{2} - 4 \times 3 \times (n - 5)$
$= 144 - 12 n + 60$
$= 204 - 12 n$

For equal roots,
$D = 0$
$\Rightarrow 204 - 12 n = 0$
$\Rightarrow n = \frac{204}{12}$
$\Rightarrow n = 17$

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